115  Epsilon Shrinkage Evaluation

NONMEM 7 Technical Guide

The general shrinkage evaluation of the \(p\)th epsilon is evaluated as

\[ R(p) = 100\% \left(1 - \sqrt{\frac{ \displaystyle\sum_{i=1}^{m} S_i(p) }{ \displaystyle\sum_{i=1}^{m} N_i(p) }}\right) \tag{1}\]

summed over subjects \(i\) to \(m\), where

\[ S_i(p) = \sum_{k=1}^{M_i} \delta_{ikp} (\mathbf{y}_{ik} - \mathbf{f}_{ik}(\hat{\boldsymbol{\eta}}))' \mathbf{V}_{ik}^{-1} (\mathbf{y}_{ik} - \mathbf{f}_{ik}(\hat{\boldsymbol{\eta}})) \tag{2}\]

\[ N_i(p) = \sum_{k=1}^{M_i} \delta_{ikp} n_k \tag{3}\]

Where in turn

\[ n_k = \left[\sum_{j \in k} 1\right] \tag{4}\]

is the number of data points of subject \(i\) that belong to correlated data cluster \(k\),

\[ \mathbf{y}_{ik} = \{y_{ij},\ j \in k\} \tag{5}\]

the vector of a subset of data points of subject \(i\), which belong to correlated data cluster \(k\) of subject \(i\). Also,

\[ \mathbf{H}'_{ik} = \{h'_{ijp},\ j \in k,\ p = 1\ \text{to}\ n\} \tag{6}\]

Where \(n\) = number of epsilons.

\[ \delta_{ikp} = \delta\left(\sum_{j \in k} h^2_{ijp}\right) \tag{7}\]

\[ \delta(x) = \begin{cases} 0 & \text{for } x = 0 \\ 1 & \text{otherwise} \end{cases} \tag{8}\]

That is, the delta function is 0 if for the \(p\)th epsilon no data point in \(k\) contributes to its evaluation. This assures that the epsilon shrinkage evaluation includes only residual terms that relate to that epsilon. For example, it assures that epsilons involved only in PK data incorporate only PK data residuals, etc. And,

\[ \mathbf{V}_{ik} = \mathbf{H}'_{ik} \boldsymbol{\Sigma} \mathbf{H}_{ik} \tag{9}\]

is the residual variance matrix to subject \(i\), data cluster \(k\).

If all data points in a subject are independent, then each data point cluster \(k\) contains only its own data point \(k\), so \(n_k = 1\) for all \(k = 1\) to \(M_i\), the number of data points to subject \(i\), and the above vectors and matrices are scalar quantities